由於元素釋放法可以不受關聯條件的限制，以適用區域的節點資料配合MLS內插觀念推導而出的形狀函數與近似位移函數，不具有Kronecker delta的性質。因此，在界面上施加Lagrange multipliers來滿足不同材料界面的位移(displacements)與曳力(tractions)連續條件，以確保界面上的連續性。
對界面附近節點資料的處理，本文引入簡易的節點選取修正方法，並配合無網格的特性，在界面附近區域任意增加或移動節點，來改善求解的精度。因此，當位移梯度(應變場)越過不同材料界面時，元素釋放法能夠解出具高階連續的變化場，使得位移梯度能滿足跳躍(jump)現象，故在處理非均質材料的問題時，元素釋放法可容易履行應力與應變場的組構定律(constitutive laws)。

Since the EF method was not restricted by connectivity, the shape functions and approximate displacement functions, which deduced from the nodal data within the adapted area and the MLS concept, had no characteristics of Kronecker delta. To ensure the continuing conditions on the interfaces, the Lagrangian multipliers were enforced on the interfaces such that the displacements and the tractions between interfaces of materials were fulfilled.
A straightforward modification to the EF method was introduced in this research that enabled the EF method to solve problems involving material discontinuities. Therefore, as the displacement gradients crossing the interfaces of materials, the high-order continuous variation field could be obtained with the EF method such that the jump of displacement gradients could be simulated. EF method satisfied the constitutive laws of stress and strain in dealing with the problems of inhomogeneous materials.

[1]. Reddy, J. N., An Introduction to Finite Element Method, McGraw-Hill, Singapore, (1993).
[2]. Logan, Daryl. L., A First Course in the Finite Element Method, PWS, Boston, (1993).
[3]. Shepard, D., “A Tow-dimensional Interpolation Function for Irregularly Spaced Points,” Proceedings A. C. M. National Conference, pp. 517-524 (1968).
[4]. Nayroles, B., G. Touzot and P. Villon, “Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Element,”Computational Mechanics, Vol. 10, pp. 307-318 (1992).
[5]. Belytschko, T., Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, “Meshless Methods: An Overview and Recent Developments,” Computer Methods in Applied Mechanics and Engineering, Vol. 139, pp. 3-47 (1996).
[6]. Belytschko, T., Y. Y. Lu and L. Gu, “Element-Free Galerkin Method,” International Journal for Numerical Methods in Engineering, Vol. 37, pp. 229-256 (1994).
[7]. Lu., Y. Y., T. Belytschko and L. Gu, “A New Implementation of the Element-Free Galerkin Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 113, pp. 397-414 (1994).
[8]. Krysl, P. and T. Belytschko, “Element-Free Galerkin Method: Convergence of the Continuous and Discontinuous Shape Functions,” Computer Methods in Applied Mechanics and Engineering, Vol. 148, 257-277 (1997).
[9]. Krongauz, Y. and T. Belytschko, “Enforcement of Essential Boundary Conditions in Meshless Approximations Using Finite Element,” Computer Methods in Applied Mechanics and Engineering, Vol. 131, pp. 133-145 (1996).
[10]. Mukherjee, Y. X. and S. Mukherjee, “On Boundary Conditions in the Element-Free Galerkin Method,” Computational Mechanics, Vol. 19, pp. 264-270 (1997).
[11]. Mackinnon, R. J. and G. F. Carey, “Treatment of Material Discontinuities in Finite Element Computations,” International Journal for Numerical Method in Engineering, Vol. 24, pp. 397-417 (1987).
[12]. Cordes, L. and B. Moran, “Treament of Material Discontinuity in the Element-Free Galerkin Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 139, pp. 75-89 (1996).
[13]. Chu, Y. A. and B. Moran, “A Computational Model for Nucleation of Solid-Solid Phase Transformations,”Modelling Simulation Materials Science Engineering, Vol. 3, pp. 455-471 (1995).
[14]. Krongauz, Y. and T. Belytschko, “EFG Approximation with Discontinuous Derivatives,” International Journal for Numerical Methods in Engineering, Vol. 41, pp. 1215-1233 (1998).
[15]. Chung, H. J. and T. Belytschko, “An Error Estimate in the EFG Method,” Computational Mechanics, Vol. 21, pp. 91-100 (1998).
[16]. Dolbow, J. and T. Belytschko, “Numerical Integration of the Galerkin Weak Form in Meshfree methods,” Computational Mechanics, Vol. 23, pp. 219-230 (1999).
[17]. Teodorico, G. T., Fatigue Crack Propagation Modeling Using the Element-Free Galerkin Method, EVANSTON, Illinois (1994).
[18]. Nemat-Nasser, S. and M. Hori, “Micromechanics: Overall Properties of Heterogeneous Materials,” North-Holland, Amsterdam (1993).
[19]. 錢偉長，「廣義變分原理」，亞東書局 (1988)。
[20]. 錢偉長，「變分法與有限元素法」，亞東書局 (1989)。
[21]. 張福范，「複合材料層間應力」，高等教育出版社，中國大陸 (1986)。
[22]. 朱伯芳，「有限單元法原理與應用」，中國水利水電出版社，中國大陸(1998)第二版。
[23]. 吳振瑞，「元素釋放法之邊界處理」，碩士論文，國立中央大學土木工程研究所，中壢 (1999)。